Theoretical Asymptotic Analysis Research Articles

Numerical model development for critical region flow and thermodynamic transitions of CO2 fluids in microchannel

Dramatic changes of thermophysical properties in the vicinity of critical point brings significant accelerating effect on the thermodynamic transitions of fluid. This study performs a confined geometric design with CO2 fluids passing through the critical zone in microchannel. Such a problem formulation combines the compressible Navier-Stokes equations with different property formulations to obtain critical anomaly effects at complex critical transport coefficients. Three equations of state (van der Waals (vdW), Redlich-Kwong (R-K) and Peng-Robinson (P-R) state equations) and their corresponding thermodynamic relations are derived in solving the governing equations of flow and heat transfer equations. Basic results are found to be consistent with theoretical asymptotic analysis, with linearly curves of pressure and a sudden shift of density (425 kg/m3–502 kg/m3) and velocity (0.82 m/s to 0.97 m/s) profiles when the microchannel flow across the critical region. The cascade effect between weakened radial thermal conductivity (λ ∼ (ρ – 1)-1/2, cp ∼ (ρ - 1)−2) and enhanced fluid expansion (βp ∼ (ρ - 1)−2) in the critical region is also shown to cause a sharp negative transition in temperature. In addition, the simulations demonstrate a qualitative comparison for three state equations, and the distinctions in quantitative terms (negative temperature value are −0.1 (vdW), −0.17 (R-K), and − 0.23(P-R)) are found to originate from stronger divergence characteristics (the specific heat cp and thermal expansion coefficients βp) in P-R gases than vdW and R-K gases.

일방향 주기적 진동상태인 원통 내부 유체의 유동 및 열전달에 대한 점근해석

A study is made of time-dependent flow and heat transfer of an incompressible Boussinesq fluid in an oscillating infinite cylinder along the horizontal axis under micro-gravity environment. The flow and heat transfer are initiated by forcing a periodic oscillation on the cylinder containing an incompressible fluid and, simultaneously, imposing an abrupt change of wall temperature. For the asymptotic analysis, Boussinesq approximation is adopted. Assuming the approximation of high Reynolds number (R<SUB>e</SUB> ≫ 1), small magnitude of thermal expansion of the fluid (ϵ ≪ 1) and moderate amplitude of the oscillation (δ), a theoretical asymptotic analysis has been performed on flow and temperature field. The results clarify the reason why heat transfer could be agumented along the axis of oscillation. The predictions are consistent with the previous numerical results.

Coulomb Gases Under Constraint: Some Theoretical and Numerical Results

We consider Coulomb gas models for which the empirical measure typically concentrates, when the number of particles becomes large, on an equilibrium measure minimizing an electrostatic energy. We study the behavior when the gas is conditioned on a rare event. We first show that the special case of quadratic confinement and linear constraint is exactly solvable due to a remarkable factorization, and that the conditioning has then the simple effect of shifting the cloud of particles without deformation. To address more general cases, we perform a theoretical asymptotic analysis relying on a large deviations technique known as the Gibbs conditioning principle. The technical part amounts to establishing that the conditioning ensemble is an I-continuity set of the energy. This leads to characterizing the conditioned equilibrium measure as the solution of a modified variational problem. For simplicity, we focus on linear statistics and on quadratic statistics constraints. Finally, we numerically illustrate our predictions and explore cases in which no explicit solution is known. For this, we use a Generalized Hybrid Monte Carlo algorithm for sampling from the conditioned distribution for a finite but large system.

Coupled isogeometric Finite Element Method and Hierarchical Genetic Strategy with balanced accuracy for solving optimization inverse problem

The liquid fossil fuel reservoir exploitation problem (LFFEP) has not only economical signification but also strong natural environment impact. When the hydraulic fracturing technique is considered from the mathematical point of view it can be formulated as an optimization inverse problem, where we try to find optimal locations of pumps and sinks to maximize the amount of the oil extracted and to minimize the contamination of the groundwater. In the paper, we present combined solver consisting of the Hierarchical Genetic Strategy (HGS) with variable accuracy for solving optimization problem and isogeometric finite element method (IGA-FEM) with different mesh size for modeling a non-stationary flow of the non-linear fluid in heterogeneous media. The algorithm was tested and compared with the strategy using Simple Genetic Algorithm (SGA) as optimization algorithm and the same IGA-FEM solver for solving a direct problem. Additionally, a parallel algorithm for non-stationary simulations with isogeometric L2 projections is discussed and preliminarily assessed for reducing the computational cost of the solvers consisting of genetic algorithm and IGA-FEM algorithm. The theoretical asymptotic analysis which shows the correctness of algorithm and allows to estimate computational costs of the strategy is also presented.

A numerical approach for the establishment of strain gradient constitutive relations in periodic heterogeneous materials

In this work, we developed a new numerical approach for the establishment of strain gradient constitutive equations in heterogeneous materials with periodic microstructure through homogenization over a unit cell. To this end, we considered a periodic unit cell of finite size, including different homogenous phases and subjected to gradient remote loading. The local elasticity problem was solved by combining the theoretical asymptotic analysis with the Fast Fourier Transform (FFT) method. The asymptotic analysis allows the establishment of differential equations of different orders. These equations were resolved by using the FFT method. By using the homogenization procedure proposed in Li (2011a,b), intrinsic strain gradient constitutive relations can be constructed. This approach fulfils the severe exigency on accuracy of the high-order homogenization. Numerical simulations on examples with 1D and 2D microstructures showed that this approach is highly accurate and efficient. Discussions were completed on the mechanical implications of the strain gradient constitutive laws through the obtained numerical results.

Non Data-Aided Estimation of the Modulation Index of Continuous Phase Modulations

In this paper, a new non data-aided estimator of the modulation index of continuous phase modulated (CPM) signals is proposed. It is based on the observation that the inverse of the index is the smallest positive real number a CPM signal should be raised to in order to generate a sinusoid of period 2T, where T is the symbol period. The asymptotic behavior of the estimator is studied. If N is the sample size, the estimation error is shown to converge to a non-Gaussian distribution at a rate of 1/N. Simulation results sustain the conclusions of the theoretical asymptotic analysis.

Performance analysis of blind carrier frequency offset estimators for noncircular transmissions through frequency-selective channels

This paper deals with the problem of blind estimation of the carrier frequency offset of a linearly modulated noncircular transmission through an unknown frequency-selective channel. A frequency estimator is developed based on the unique conjugate cyclic frequency of the received signal, which is equal to twice the frequency offset. Consistency and asymptotic normality of the frequency estimator together with a closed-form expression for its asymptotic variance are also established. The closed-form expression of the asymptotic variance enables analysis of the performance of the proposed frequency offset estimator as a function of the number of estimated cyclic correlation coefficients used. It is shown that optimum is obtained if the number of correlation coefficients taken into account coincides with the degree of the channel. Numerical simulations are provided and confirm the conclusion of the theoretical asymptotic analysis.

On propagation through long step tapers

In integrated optics, tapered waveguides usually possess step changes in waveguide width or other dimensions, due to lithographic limitations. We present a theoretical asymptotic analysis of the effect of these steps when compared with the ideal smooth taper. We present some figures for the size of the effect, conclude that the effect is significant in certain situation and that making the taper too long will reduce its performance, or in the worst instance cause the taper to function as badly as a butt join.

Is More Information a Good Thing? Bias Nonmonotonicity in Stochastic Difference Equations

It is shown that the bias of estimated parameters in autoregressive models can increase as the sample size grows. This bias is also a nonmonotonic function of the largest autoregressive root, contrary to what asymptotic approximations had indicated so far in the literature. These unusual results are due to the effect of the initial sample observations that are typically neglected in theoretical asymptotic analysis, in spite of their empirical relevance. Implications for practical economic modelling are considered, including a comparison of the likely inaccuracies of parameter estimates in alternative models based on competing macroeconomic theories.

Asymptotic Analysis of Intraparticle Diffusion in GAC Batch Reactors

The homogeneous surface-diffusion model (HSDM) and the pore-surface-diffusion model (PSDM) for adsorption on granular activated carbon in a completely mixed batch reactor are examined in the near-equilibrium regime. In this asymptotic regime, linearized model equations yield analytic solutions for the measurable rate of adsorbate uptake involving effective diffusion coefficients. The theoretical asymptotic analysis is verified by numerical simulation of the complete equations. A basis is therefore provided for a method of determining diffusion coefficients from batch experiments that is simpler, more flexible, and more robust than those presently used. The present work also clarifies differences in the qualitative effects of pore and surface diffusion and suggests a preliminary graphical criterion for assessing directly from experimental data the possible importance of surface diffusion relative to other possible intraparticle transport mechanisms. A method for determining both pore- and surface-diffusion coefficients when the pore-surface-diffusion model is appropriate and the appropriate data available are described. The proposed approach is illustrated by the analysis of two data sets available in the literature, and the resulting estimates are compared with those obtained by a more conventional method.

Asymptotic modelling of the transient regime in heat conduction in solids with general geometry

This paper proposes a mathematical model for predicting the evolution of mean and thermal centre temperatures of generally convex solid bodies subjected to cooling or heating processes without a phase change. To do so, a theoretical asymptotic analysis is made under boundary conditions of the third kind of the Fourier number, valid for the aforementioned temperatures. Similar asymptotic functions were obtained in the ( Bi → 0 and Bi → ∞) limits. This enables a methodology to be established in the direct process time modelling based on its evolution as a function limited between the asymptotic values. The study is carried out in stages, starting with solids with elemental geometries up to those with general geometries. A specific, simple, easily programmed model is proposed for evaluating the said variable.

Convection patterns in a triangular domain

The phenomenon of natural convection in trapezoidal enclosures where upper and lower walls are not parallel, in particular a triangular geometry, is re-examined over a parameter domain in which the aspect ratio of the enclosure ranges from 0.1 to 1.0, the Rayleigh number varies between 10 2 and 10 5, and the Prandtl numbers correspond to air and water. The time dependent, two-dimensional non-linear problem is solved numerically and the results confirmed by independent methods. It is found that the numerical experiments verify the flow features that are known from theoretical asymptotic analysis of this problem (valid for shallow spaces) only over a certain range of the parametric domain, and that this solution breaks down as the Rayleigh number is increased beyond 3 × 10 3, when a bifurcation takes place and a new steady state is achieved, that of a multi-cellular type. The transient numerical experiments show that in the present parametric domain the single-cell circulation is not stable with respect to the Bénard instability one expects in fluid layers heated from below. This result is supported by recent experimental and numerical results reported on this problem.